Applications of sequential Bayesian Inference in the geosciences, such as atmosphere, ocean, atmospheric chemistry, and land-surface, are characterised by high dimensions, nonlinearity, and complex relations between system variables. While Gaussian-based approximations such as Ensemble Kalman Filters and Smoothers and global variational methods have been used quite extensively in this field, numerous problems ask for methods that can handle strong nonlinearities. In this talk I will discuss recent progress using particle filters.

Three main areas of active research in particle filtering can be distinguished, exploring localisation, exploring proposal densities, and exploring (optimal) transportation (and mergers of these ideas are on the horizon). In localisation the idea is to split the high-dimensional problem in several smaller problems that then need to be stitched together in a smart way. The first approximate applications of this methodology have just made it to weather prediction, showing the exponentially fast developments here. However, the ‘stitching’ problem remains outstanding. The proposal density methodology discussed next might be fruitful to incorporate here.

In the proposal density approach one tries to evolve states in state space such that they obtain very similar weights in the particle filter. Challenges are, of course, the huge dimensions, but these also provide opportunities via the existence of typical sets, which lead to preferred parts of state space for the particles. Recent attempts to exploit typical sets will be discussed.

Finally, we will discuss recent progress in (optimal) transportation. The idea here is that a set of prior particles has to be transformed to a set of posterior particles. This is an old problem in optimal transportation. However, the optimality condition poses unnecessary constraints, and by relaxing the optimality constraint we are able to formulate new efficient methods. Specifically, by iteratively minimising the relative entropy between the probability density of the prior particles and the posterior a sequence of transformations emerges for each particle that seems to be tractable even for very high dimensional spaces. A new idea is to explore localisation to obtain a more accurate description of the target posterior, but without the stitching issues mentioned above.

So far, model reduction techniques, emulation, and machine learning techniques have been unsuccessful for these high-dimensional state estimation problems, but I’m keen to further understand the possibilities and limitations.